CONSTITUTIVE MODELS FOR DUCTILE SOLIDS REINFORCED BY RIGID SPHEROIDALINCLUSIONS

We study the effective constitutive response of composite materials ma de of rigid spheroidal inclusions dispersed in a ductile matrix phase. Given a general convex potential characterizing the plastic (in the c ontext of J2-deformation theory) behavior of the isotropic matrix, we derive expressions for the corresponding effective potentials of the r igidly reinforced composites, under general loading conditions. The de rivation of the effective potentials for the nonlinear composites is b ased on a variational procedure developed recently by Ponte Castaneda (1991a, J. Mech. Phys. Solids 39, 45-71). We consider two classes of c omposites. In the first class, the spheroidal inclusions are aligned, resulting in overall transversely isotropic symmetry for the composite . In the second class, the inclusions are randomly oriented, and thus the composite is macroscopically isotropic. The effective response of composites with aligned inclusions depends on both the orientation of the loading relative to the inclusions and on the inclusion concentrat ion and shape. Comparing the strengthening effects of rigid oblate and prolate spheroids, we find that prolate spheroids give rise to stiffe r effective response under axisymmetric (relative to the axis of trans verse isotropy) loading, while oblate spheroids provide greater reinfo rcement for materials loaded in transverse shear. On the other hand, n early spherical (slightly prolate) spheroids are most effective in str engthening the composite under longitudinal shear. Thus, the optimal s hape for strengthening composites with aligned inclusions depends stro ngly on the loading mode. Alternatively, the properties of composites with randomly oriented spheroidal inclusions, being isotropic, depend only on the concentration and shape of the inclusions. We find that bo th oblate and prolate inclusions lead to significant strengthening for this class of composites.